#278387

order	512
nilpotency class	3
isoclinism	isoclinic class of #278379
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,13)(5,15)(6,10)(7,11)(9,14)(12,16)(17,18,21,24,22,26,20,25)(19,23,27,32,29,30,28,31);(1,3,5,11,6,12,14,4)(2,7,9,8,10,13,15,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,31)(26,30);(18,24)(19,27)(20,21)(23,30)(25,26)(28,29);(1,4,14,12,6,11,5,3)(2,8,15,7,10,16,9,13)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185583);